Well-Ordered Principle: Every non-empty set of positive integers has a least element. (This is equivalent to induction.)

Non-Triviality: 0≠1{\displaystyle 0\neq 1}. *This is actually unnecessary to have as an axiom, since it can be easily be proven that 0≠1{\displaystyle 0\neq 1}. Proof: Assume 0=1{\displaystyle 0=1}. There exists a positive integer a{\displaystyle a} such that a{\displaystyle a} is a member of the positive integers. Then, a×0=a×1{\displaystyle a\times 0=a\times 1} Therefore, 0=a{\displaystyle 0=a} However, since thricotomy states that every positive integer is either equal to 0, positive, or negative, there is a contradiction such that a is both 0 and a positive integer. Therefore, 0≠1{\displaystyle 0\neq 1}. This simple proof provides a more powerful system since less has to be assumed.